Theorem prnz 4710

Description: A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)

Hypothesis

Ref	Expression
prnz.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
prnz	⊢ {𝐴, 𝐵} ≠ ∅

Proof of Theorem prnz

Step	Hyp	Ref	Expression
1		prnz.1	. . 3 ⊢ 𝐴 ∈ V
2	1	prid1 4695	. 2 ⊢ 𝐴 ∈ {𝐴, 𝐵}
3	2	ne0ii 4268	1 ⊢ {𝐴, 𝐵} ≠ ∅

Syntax hints:   ∈ wcel 2108   ≠ wne 2942  Vcvv 3422  ∅c0 4253  {cpr 4560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-sn 4559  df-pr 4561

This theorem is referenced by:  opnz  5382  propssopi  5416  fiint  9021  wilthlem2  26123  upgrbi  27366  wlkvtxiedg  27894  shincli  29625  chincli  29723  spr0nelg  44816  sprvalpwn0  44823